Topic

# Metric tensor

About: Metric tensor is a research topic. Over the lifetime, 330 publications have been published within this topic receiving 6290 citations.

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30 Nov 1993

TL;DR: In this article, the authors propose a generalized Lagrange space for embedding bundles of vector bundles, based on the generalized Einstein-Yang-Mills Equations (GSE) and generalized Lagrange Spaces (GLn).

Abstract: I. Fibre Bundles, General Theory. II. Connections in Fibre Bundles. III. Geometry of the Total Space of a Vector Bundle. IV. Geometrical Theory of Embeddings of Vector Bundles. V. Einstein Equations. VI. Generalized Einstein--Yang--Mills Equations. VII. Geometry of the Total Space of a Tangent Bundle. VIII. Finsler Spaces. IX. Lagrange Spaces. X. Generalized Lagrange Space. XI. Applications of the GLn Spaces with the Metric Tensor e2sigma(x,y)gammaij(x,y). XII. Relativistic Geometrical Optics. XIII. Geometry of Time Dependent Lagrangians. Bibliography. Index.

575 citations

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TL;DR: In this article, the Lattman metric is shown to be a scalar multiple of the metric tensor which arises naturally from the Jacobi formulation of the action principle for spherical tops.

Abstract: New sets of variables are studied which should lead to significantly improved numerical stability and efficiency for computer stimulation studies of rigid classical molecules. The search for these new variables is made by finding variables which lead to the simplest (i.e. euclidean) expressions for the metric tensor of orientation space. It is shown that the intuitively defined Lattman metric [1] is a scalar multiple of the metric tensor which arises naturally from the Jacobi formulation of the action principle for spherical tops. It is then shown that Euler's quaternion parameters lead to a euclidean form for the orientation metric. These parameters lead to many associated simplifications in the equations of motion of classical rigid bodies including the removal of singularities and spurious behaviour near θ = 0. It is felt that these benefits will translate into increased accuracy and efficiency both for numerical integration of the equations of motion and for performing Monte Carlo integrations of phas...

392 citations

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TL;DR: In this paper, the Lagrangian for interacting nonrelativistic particles can be coupled to an external gauge field and metric tensor in a way that exhibits a non-relative version of general coordinate invariance.

351 citations

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TL;DR: The concept of space-time representation in the brain is redefined using tensor network theory and the cerebellum acts as a predictive motor space- time metric which allows the establishment of coincidences of goal-directed movements of limbs inspace-time with external targets.

327 citations

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15 May 2004TL;DR: This work introduces principal geodesic analysis, a generalization of principal component analysis, to symmetric spaces and applies it to the computation of the variability of diffusion tensor data, and develops methods for producing statistics, namely averages and modes of variation, in this space.

Abstract: Diffusion tensor magnetic resonance imaging (DT-MRI) is emerging as an important tool in medical image analysis of the brain. However, relatively little work has been done on producing statistics of diffusion tensors. A main difficulty is that the space of diffusion tensors, i.e., the space of symmetric, positive-definite matrices, does not form a vector space. Therefore, standard linear statistical techniques do not apply. We show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. In our previous work we introduced principal geodesic analysis, a generalization of principal component analysis, to compute the modes of variation of data in Lie groups. In this work we expand the method of principal geodesic analysis to symmetric spaces and apply it to the computation of the variability of diffusion tensor data. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease.

296 citations